Monday, September 16, 2024

Insomnia: van Aubel's Theorem

I woke up in the middle of the night and stumbled upon this post in mathstodon about a simple theorem in Euclidean geometry. So I thought it was a good time to dust off my Geogebra app.

Van Aubel's theorem states that if you start with a convex quadrilateral and construct a square on each side, externally to the quadrilateral, then the line segments connecting the centers of opposite squares will be equal in length and perpendicular to each other.

The following images provide a synthetic proof of this theorem. The line segments mentioned are colored in green and red.

The green and red circumference cross on point O1, which is the center of the blue circumference. By construction, the blue circumference contains points M1 and K1. Hence the red and green segments are of equal length.

The purple line is the perpendiculat to the green line that passes through P1. As can be seen, the purple line contains the red segment. Hence the green and red segments are perpendicular.

This concludes the proof of this theorem.

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